Dmitrii Konstantinovich Faddeev

(on his eightieth birthday)

Dmitrii Konstantinovich Faddeev, Corresponding Member of the Academy
of Sciences of the USSR, celebrated his eightieth birthday in 1987.

Faddeev was born into the family of a Petersburg engineer on 17 (30)
June 1907 in Yukhnov, in the district of Smolensk. From 1923, when he enrolled
as a student of mathematics, until today all Faddeev's varied activity
has been closely linked with the University of Leningrad, where he is now
one of the oldest professors of mathematics, and where for many years he
held the chair of higher algebra and number theory, and where he has been
Dean of the Faculty of Mathematics and Mechanics. His links with the Academy
of Sciences of the USSR are no less close. In 1932 he began research in
the Steklov Institute of Mathematics and Physics and when in 1934 this
institute moved from Leningrad to Moscow and became the Mathematical Institute,
and the Leningrad Division of the Steklov Institute of Mathematics of the
Academy of Sciences of the USSR (LOMI) was created, Faddeev was from the
very moment of its foundation a collaborator in LOMI and he has remained
so. In 1964 he was elected Corresponding Member of the Mathematics Section
of the Academy of Sciences of the USSR. For a long time he was in charge
of the laboratories at LOMI, and head of a powerful group of well-known
algebraists and number theorists; here for some decades he ran the algebra
seminar, which in fact was spread over many cities, which brought together
colleagues from LOMI and the University, and which is widely known throughout
the whole country.

The range of Faddeev's interests is unusually broad—among his more than
150 papers are some on the theory of functions, on computational methods,
on probability theory, on problems of teaching mathematics at all levels.
But, of course, Faddeev is known in the first place as one of the most
outstanding algebraists of our time.

In his papers Faddeev touched on a wide range of algebraic problems.
But there are two areas in which he began his research and to which he
constantly returned. They are Diophantine equations and Galois theory.

Faddeev's very first results in Diophantine equations were remarkable.
He was able to extend significantly the class of equations of the third
and fourth degree that admit a complete solution. When he was studying,
for example, the equation x3+y3 = A, Faddeev
found estimates of the rank of the group of solutions that enabled him
to solve the equation completely for all A <= 50. Until then
it had been possible to prove only that there were non-trivial solutions
for some A. For the equation x4+Ay4 = ±1
he proved that there is at most one non-trivial solution; this corresponds
to the basic unit of a certain purely imaginary field of algebraic numbers
of the fourth degree and exists only when the basic unit is trinomial.

Faddeev returned again and again to the study of Diophantine equations
in the years that followed. In [66], [64], [58] he studied the group of
classes of divisors of zero degree for a Fermat curve of degrees 4, 5,
7 and he proved that it is finite: from that there followed very powerful
results for the equations themselves. For example, there are only finitely
many quadratic fields in which the equation x4+y4 = 1 has a non-trivial
solution, and in each of these fields there are only finitely many solutions.
Until then there had been no results of this kind.

Faddeev made a significant contribution to Galois theory. He was especially
interested in the inverse problem: the construction of extensions (all
or at least one) of a given field with a prescribed Galois group. In his
earlier papers [I], [5], [9], [12], [17], he had solved this problem completely
for small groups (subgroups of the group of permutations of three or four
elements, metacyclic transitive groups of permutations of a prime number
of elements, groups of quaternions and quaternion units). For this he used
a beautiful geometric method: the unknown field is interpreted as a subset
of a certain vector space, on which the action of a Galois group is described
quite simply. Many of the results obtained here are presented in an elegant
geometrical formulation. For example, extensions of the field of rational
numbers with a quaternion Galois group prove to be closely connected with
triples of mutually orthogonal vectors with rational components in three-dimensional
Euclidean space.

Howver, similar arguments proved inadequate for further progress. In
a remarkable paper of 1944 ([15], a joint paper with B.N. Delone) Faddeev
began a systematic study of a problem which is much more general than the
inverse problem—the problem of Galois embedding: to embed a given Galois
extension in a wider field with a given Galois group and with a given epimorphism
of this group onto the Galois group of the original extension. Here he
stated a necessary decidability condition for the embedding problem —the
condition of compatibility, it consists in the additive decidability of
the problem (it demands of the solution that it be a vector space on which
a group of operators acts, properly compatible with the Galois group of
the embedded field and no multiplication at all is required). It was shown
there too that compatibility is sufficient for embeddability if the kernal
of the embedding problem under consideration is a cyclic group of odd order.
Moreover, Faddeev proved in the same paper that if the embedding problem
with an Abelian kernel for number fields admits a solution that is a Galois
algebra, then there is a field that solves it.

The compatibility condition can be formulated in various ways (some
of them are stated in the same paper). It turned out to be so deep and
so close to the sufficient condition for embedding that H. Hasse (who discovered
it four years later for a special case) assumed it and tried to prove that
is is a sufficient condition for embedding (at least in the case of an
Abelian kernel). This error was made by other eminent mathematicians. However,
as follows from the examples of Shafarevich and Faddeev, Hasse's conjecture
is false. In Faddeev's example [28] the kernel of the embedding problem
is the cyclic group of order 8. A supplementary necessary and sufficient
condition for embedding in this case can be found in the paper [57], written
jointly with R.A. Shmidt. For an arbitrary Abelian kernel a supplementary
condition for embedding was found by A.V. Yakovlev, a pupil of Faddeev.

Faddeev's work on Galois theory proved to be very influential in the
further development of this theory. In particular, it (together with Faddeev's
results on cohomology of groups) played an essential role in Shafarevich's
proof of the decidability of the inverse problem of Galois theory for soluble
groups and number fields.

In 1947 Faddeev published a paper [19] in which he defined (at the same
time as the American mathematicians Eilenberg and MacLane but independently
of them) cohomology groups for groups. This created a powerful tool for
use in different fields of algebra, especially in Galois theory: cohomologies
were an ideal device for studying problems of embedding of fields, for
constructing extensions with soluble groups, and for other problems connected
with Galois theory and applications of it. In later papers Faddeev proved
many important theorems in homological algebra: of particular interest
to him was the link between the cohomologies of a group and a subgroup
of it. Very important are his joint papers [36] and [52] with Borevich,
in which not only was there given for the first time a systematic exposition
of the theory of cohomology of groups, but new interesting results were
obtained. In particular, a proof was given of the analogue of the Krull-Schmidt
theorem for operator modules over complete local rings, minimal projective
resolutions were constructed, and the existence of finitely many pairwise
non-isomorphic indecomposable modules of p-adic representations of a non-cyclic
p-group
was proved.

In [24] and [38] the homological apparatus was effectively used in the
study of simple algebras over the field of algebraic functions of one variable.
A theory was constructed which in many ways is analogous to the theory
of algebras over fields of algebraic numbers. In particular, in [24] the
Brauer group of the field of rational functions over a field of characteristic
0 was calculated. This result was considerably ahead of its time, and was
only proved again 17 years later by the American mathematicians Auslander
and Brumer.

At the end of the 50's Faddeev turned to problems in the theory of integral
representations. His numerous lectures and reports drew attention to this
group of problems and contributed to the intense development of work on
them. In a long article [77] the strategy was laid down for work in the
field of integral representations. In this paper he put forward a multiplicative
theory of lattices in algebras over the quotient field of a Dedekind ring;
with the help of this theory it was possible to isolate six different levels
for classifying these lattices. The most precise level was the classification
up to a similitude; the crudest was the coincidence of coefficient rings.
In some cases different levels of classification may coincide; for example,
for commutative rings there are only three levels left (besides those already
listed up to a local similitude). In this way the general problem of classification
is divided into a series of steps, each of which requires an independent
approach.

In a series of papers ([79] and jointly with Borevich [78], [80]) he
applied this programme to specific rings. In [79] he studied cubic Z-rings,
which were of special interest, since they are closely connected with earlier
research by Faddeev and Delone into cubic irrationalities. In the other
papers, written jointly with Borevich, they studied representations of
orders with a cyclic index in the maximal order. They proved that every
finitely generated torsion-free module over such an order decomposes into
a direct sum of ideals of the order and gave a complete set of invariants
that specify the module. Bass studied the same class of rings from a different
angle;

using the results of [78] he showed that a cyclic index is not only
a sufficient but also a necessary condition for the decomposability of
a torsion- free module into a sum of ideals.

Faddeev's contribution to the theory of representations was not confined
to his own results. Under his influence a powerful school was founded in
the USSR which produced many outstanding results.

The papers [115], [119], [120], [122] treat another aspect of the theory
of representations. They contain a beautiful classification of complex
representations for classical groups over finite fields in terms of Green's
multiplication. For the full linear group the so-called simple representations
are distinguished, which are not contained in products of representations
of full linear groups of lower orders. The irreducible components of a
power of a simple representation are called primary; they correspond to
representations of the symmetric group of order equal to the degree. An
arbitrary representation is uniquely represented in the form of a product
of primaries. For the full affine group any irreducible representation
is obtained by multiplying the so-called "pivotal" representations by arbitrary
representations of the full linear group. The pivotal representation in
every dimension is unique and is induced from the one-dimensional non-singular
representation of the group of upper unitriangular matrices.

Faddeev is one of the leading experts in numerical methods of linear
algebra. His work in this field was mostly done in collaboration with V.N.
Faddeeva, and played an important role in establishing and developing this
science. The survey lectures they gave at many all-union and international
conferences, their survey articles on numerical methods of algebra ([48],
[118], [124], [125]), their monograph [63] Vychislitel'nye metody lineinoi
algebry (Numerical methods of linear algebra), published in 1960 and in
revised form in 1963, played a great part both in the training of computational
mathematicians and in the development of numerical methods in linear algebra.
The monographs of Faddeev and Faddeeva won world-wide recognition, were
translated into many languages, and are still relevant today. They contain
a large number of methods and algorithms for solving algebraic problems,
a profound analysis of the principles behind their construction, and a
systemization of them. Many methods were further developed. Characteristic
of Faddeev in his striking sense for what is new and promising. His talks,
monographs, and articles are not only excellent guides through the labyrinth
of published material, but they contain many ideas giving food for creative
thought.

Most of Faddeev's own research is basically concerned with the stability
of the numerical solution of systems of linear algebraic equations and
estimation of the results of calculations. The stability of the solution
under a variation of the input data (the conditioning of the system) and
an investigation of its quantitative characteristics are considered in
[54], [68], and [90]. Here in particular we can clearly see the influence
of Turing's condition number on the estimate of the error of the solution
when the elements of a matrix are independent random numbers with a common
small dispersion; in terms of the singular decomposition of the matrix
the analysis of the system is carried out on the condition that makes it
possible to clarify what information an ill-conditioned system carries
within itself; here is posed and answered the problem of finding two diagonal
matrices D1 and D2 yielding the minimum
number of the Turing condition for the matrix D1AD2.

A new approach to estimating the complete error in the solution of a
problem in finite calculation was developed in [96], [98], [112], [126].
In particular, within the limits of the linearized theory of errors there
is a proposal to describe the natural domain of solutions and to replace
an estimate of its norm by an estimate of elliptic norms specially chosen
for the given problem and closely approximating the natural norms. The
construction of such elliptic norms is achieved by using the device of
companion matrices. This method imitates the probability estimates of the
linear probability theory of errors. The link between companion and correlation
matrices is studied, and also the transition from norms in which the input
data are given to elliptic norms, and an estimate of the loss of information
occurring during this transition.

The papers [91] and [93] deal with the solution of linear systems of
a general form, where from geometrical positions light is thrown on questions
connected with the solution of systems with rectangular matrices. An algorithm
is proposed for factorizing an arbitrary matrix into a left trapezoidal
matrix with a certain prescribed ordering of elements and a matrix of orthonormalized
columns. A link is established between the singular numbers of the matrix
of the system to be solved and the diagonal elements of the left trapezoidal
matrix. On the basis of the above mentioned factorization, numerically
stable algorithms are proposed for solving systems with rectangular matrices,
and also an algorithm for analysing ill-conditioned systems.

In [113] a new concept is suggested for estimating the quality of the
results of the numerical solution of linear-algebraic systems depending
on the quality of the specification of the input data. Three situations
are considered:

scholarly (the input data are given precisely), regular (possible variations
of the input data are far from critical), and irregular (variations of
the input data are close to or coincide with the critical problem, which
reduces to a qualitative variation). An analysis is given of the solution
of the problem in each of these situations.

Of Faddeev's other papers we must mention [ 108] and [111]. In [ 108]
he investigates the properties of consistency and multiplicativity for
norms in spaces of polynomial forms of vectors that belong to finite-dimensional
spaces with vector norms. In [111] he proposes a method for embedding the
algebra of matrices of lower order in the algebra of matrices of higher
order, which could be used for the approximation of large problems by small
ones, both for the solution of systems and in the spectral problem of matrices.

Problems in the teaching of mathematics in schools have long been one
of the themes of Faddeev's scientific and teaching work. The book "Algebra"
by Faddeev and I.S. Sominskii appeared as early as 1951 and was reissued
in 1964, a book aimed at giving greater depth to the teaching of algebra
courses in schools. Faddeev was one of the founders and organizers of the
Physics and Mathematics Boarding School at Leningrad State University.

A distinctive feature of Faddeev's approach to the teaching of mathematics
in schools is his clearly expressed and logically developed chain of ideas
in the courses he devised on algebra and basic analysis. The essence of
his ideas is expressed in his own words: "The elementary part of mathematics
is the simplest of the sciences, since its object is a study of the crudest
aspects of reality". Faddeev then goes on to subtle mathematical ideas
of continuity, of limit, of smoothness of a function, of a derivative,
by way of a practical, intelligent and interesting use of the concept of
a number as the result of measurement or of a rationally and appropriately
executed calculation. Faddeev gives priority to "vivid contemplation",
to awareness, to direct perception of the properties of mathematical objects,
which thus

reveal the content and basic ideas of mathematics as tools and as a
description and study of the regularity of the real world.

Faddeev formulated one of his fundamental methodological and pedagogical
principles in the preface to his book Lektsii po algebre (Lectures on algebra):

"I consider that they (that is, abstract concepts) must be introduced
to the extent that their introduction succeeds in stimulating in the students
the need to generalize or, at least, a realization that it is possible
to illustrate •sufficiently general concepts by more concrete material".
Faddeev maintained this principle in his books for the middle school "Algebra
6-8" and Elementy vysshei matematiki dlya shkol'nikov (Elements of higher
mathematics for school-children).

In Faddeev's course, mathematical formalism as an essential effective
tool and instrument of pure mathematical research comes after the corresponding
concepts and ideas have been interpreted and developed at an interesting
intuitive level. And even the second (formal) level of presentation gives
the curious and keen pupil examples of sufficiently rigorous mathematical
reasoning and proof. Thus, in Faddeev's works, directed at both pupils
and teachers, there is to be found a solution to the complicated task of
combining a general educational direction with the special training of
pupils to continue with the study of mathematics and the application of
it at a professional level.

The lines we have listed do not exhaust all aspects of Faddeev's mathematical
activity. His numerous and very brilliant papers on the theory of functions,
probability theory, and so on, are well known. His teaching over many years
at the University, his general and specialized courses of lectures, always
excellently presented, his numerous public speeches, are a splendid example
to younger mathematicians. His Kurs lektsii po algebre (Course of lectures
on algebra) and Sbornik zadach po vysshei algebre (Collection of problems
on higher algebra) written on this basis are rightly considered the best
books of their kind.

Dmitrii Konstantinovich Faddeev is a man of the broadest culture and
intelligence. He has a wide knowledge and appreciation of classical music
and is an outstanding pianist. Discussion with him of a variety of questions
is highly esteemed by all his friends, colleagues, and acquaintances.

His pupils and colleagues wish Dmitrii Konstantinovich good health,
new success in his research, and happiness.